Question

in the figure below, ( mangle1 = 3x^{circ} ) and ( mangle2=(x + 14)^{circ} ). find the angle measures.

Explanation:

Step1: Set up the equation

Since \(\angle1\) and \(\angle2\) are complementary (they form a right - angle, so \(m\angle1 + m\angle2=90^{\circ}\)). Substitute \(m\angle1 = 3x^{\circ}\) and \(m\angle2=(x + 14)^{\circ}\) into the equation: \(3x+(x + 14)=90\).

Step2: Simplify the equation

Combine like terms: \(3x+x+14 = 90\), which gives \(4x+14 = 90\).

Step3: Solve for \(x\)

Subtract 14 from both sides: \(4x=90 - 14\), so \(4x=76\). Then divide both sides by 4: \(x=\frac{76}{4}=19\).

Step4: Find \(m\angle1\)

Substitute \(x = 19\) into \(m\angle1=3x\). So \(m\angle1=3\times19 = 57^{\circ}\).

Step5: Find \(m\angle2\)

Substitute \(x = 19\) into \(m\angle2=x + 14\). So \(m\angle2=19+14=33^{\circ}\).

Answer:

\(m\angle1 = 57^{\circ}\)
\(m\angle2 = 33^{\circ}\)